How can I make an object that is already animated follow a path correctly?
Your object is not parented to the path and doesn't have any constraint. To make it move along the path, you can simply parent it to the path, then select the path, in the Graph Editor > N panel > Modifiers, delete the Generator, then in Properties panel > Data > Path Animation, give some Frames, like for example 100, then play with the Evaluation Time and put some keyframes to move your object along the path at the speed you want.